Abstract

In this paper we analyze the problem of light-matter interaction when absorptive resonances are imbedded in the material dispersion. We apply an improved approach to aluminum (Al) in the optical frequency range to investigate the impact of these resonances on the operating characteristics of Al-based nanoscale devices. Quantities such as group velocity, stored energy density, and energy velocity, normally obtained using a single resonance model [Wave Propagation and Group Velocity (Academic Press, 1960), Nat. Mater. 11, 208 (2012)], are now accurately calculated regardless of the medium adopted. We adapt the Loudon approach [Nat. Mater. 11, 208 (2012)] to media with several optical resonances and present the details of the extended model. We also show pertinent results for Al-based metal-dielectric-metal (MDM) waveguides, around spectral resonances. The model delineated here can be applied readily to any metal accurately characterized by Drude-Lorentz spectral resonance features.

© 2012 OSA

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References

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  1. Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
    [CrossRef] [PubMed]
  2. N. Engheta, “Taming light at the nanoscale,” Phys. World 23, 31–34 (2010).
  3. L. Brillouin, Wave Propagation and Group Velocity (Academic Press Inc., 1960).
  4. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, 1984).
  5. R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3(3), 233–245 (1970).
    [CrossRef]
  6. F. D. Nunes, T. C. Vasconcelos, M. T. Bezerra, and J. Weiner, “Electromagnetic energy density in dispersive and dissipative media,” J. Opt. Soc. Am. B 28(6), 1544–1552 (2011).
    [CrossRef]
  7. K. E. Oughstun and S. Shen, “Velocity of energy transport for a time-harmonic field in a multiple-resonance Lorentz medium,” J. Opt. Soc. Am. B 5(11), 2395–2398 (1988).
    [CrossRef]
  8. R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299(2-3), 309–312 (2002).
    [CrossRef]
  9. J. A. Stratton, Electromagnetic theory, 1st ed. (McGraw-Hill Book Company, 1941).
  10. D. Y. Smith, E. Shiles, and M. Inokuti, “The optical properties of metallic aluminum,” Edward D. Palik ed., in Handbook of Optical Constants of Solids (Academic Press 1998), p 369.
  11. G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
    [CrossRef] [PubMed]
  12. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
    [CrossRef] [PubMed]
  13. A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73(2), 218–224 (2006).
    [CrossRef]
  14. M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18(11), 3117–3126 (2006).
    [CrossRef]
  15. M. Mansuripur, Field, Force, Energy and Momentum in Classical Electrodynamics, 1st ed. (Bentham Science Publishers Ltd., 2011).

2012 (1)

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

2011 (1)

2010 (1)

N. Engheta, “Taming light at the nanoscale,” Phys. World 23, 31–34 (2010).

2006 (4)

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73(2), 218–224 (2006).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18(11), 3117–3126 (2006).
[CrossRef]

2002 (1)

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299(2-3), 309–312 (2002).
[CrossRef]

1988 (1)

1970 (1)

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3(3), 233–245 (1970).
[CrossRef]

Alù, A.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

Barsi, C.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[CrossRef] [PubMed]

Bezerra, M. T.

Bigelow, M. S.

A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73(2), 218–224 (2006).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18(11), 3117–3126 (2006).
[CrossRef]

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18(11), 3117–3126 (2006).
[CrossRef]

A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73(2), 218–224 (2006).
[CrossRef]

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[CrossRef] [PubMed]

Dolling, G.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Edwards, B.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

Engheta, N.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

N. Engheta, “Taming light at the nanoscale,” Phys. World 23, 31–34 (2010).

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Gehring, G. M.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[CrossRef] [PubMed]

Jarabo, S.

A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73(2), 218–224 (2006).
[CrossRef]

Kostinski, N.

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[CrossRef] [PubMed]

Lepeshkin, N. N.

A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73(2), 218–224 (2006).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18(11), 3117–3126 (2006).
[CrossRef]

Linden, S.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Loudon, R.

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3(3), 233–245 (1970).
[CrossRef]

Nunes, F. D.

Oughstun, K. E.

Ruppin, R.

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299(2-3), 309–312 (2002).
[CrossRef]

Schweinsberg, A.

A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73(2), 218–224 (2006).
[CrossRef]

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[CrossRef] [PubMed]

Shen, S.

Shin, H.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18(11), 3117–3126 (2006).
[CrossRef]

Soukoulis, C. M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Sun, Y.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

Vasconcelos, T. C.

Wegener, M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Weiner, J.

Europhys. Lett. (1)

A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, and S. Jarabo, “Observation of superluminal and slow light propagation in erbium-doped optical fiber,” Europhys. Lett. 73(2), 218–224 (2006).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. A (1)

R. Loudon, “The propagation of electromagnetic energy through an absorbing dielectric,” J. Phys. A 3(3), 233–245 (1970).
[CrossRef]

J. Phys. Condens. Matter (1)

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of a smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys. Condens. Matter 18(11), 3117–3126 (2006).
[CrossRef]

Nat. Mater. (1)

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11(3), 208–212 (2012).
[CrossRef] [PubMed]

Phys. Lett. A (1)

R. Ruppin, “Electromagnetic energy density in a dispersive and absorptive material,” Phys. Lett. A 299(2-3), 309–312 (2002).
[CrossRef]

Phys. World (1)

N. Engheta, “Taming light at the nanoscale,” Phys. World 23, 31–34 (2010).

Science (2)

G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science 312(5775), 895–897 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312(5775), 892–894 (2006).
[CrossRef] [PubMed]

Other (5)

J. A. Stratton, Electromagnetic theory, 1st ed. (McGraw-Hill Book Company, 1941).

D. Y. Smith, E. Shiles, and M. Inokuti, “The optical properties of metallic aluminum,” Edward D. Palik ed., in Handbook of Optical Constants of Solids (Academic Press 1998), p 369.

L. Brillouin, Wave Propagation and Group Velocity (Academic Press Inc., 1960).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, 1984).

M. Mansuripur, Field, Force, Energy and Momentum in Classical Electrodynamics, 1st ed. (Bentham Science Publishers Ltd., 2011).

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Figures (10)

Fig. 1
Fig. 1

Frequency dependence of the real (left panel) and imaginary (right panel) parts of Al permittivity. Experimental data from Smith et al. [9] (circles), and their fitting curves (solid lines) using Eq. (28) for left panel and Eq. (29) for right panel. Dashed curves represent the Drude model contribution alone, shown here for the sake of comparison.

Fig. 2
Fig. 2

Real (left panel) and imaginary (right panel) parts of the refractive index of aluminum calculated with multi-Lorentz approach. Dashed curves show the same quantity when only the Drude contribution is taken into account.

Fig. 3
Fig. 3

Group index for jAl calculated with the multiresonance model. Painted area highlights the spectral range Δω where Ng > 1.0. Dashed curve shows group velocity when only Drude model contribution is considered.

Fig. 4
Fig. 4

Stored energy density (left) and dissipated power (right) for interaction of electromagnetic radiation in Al calculated with multi-Lorentz approach and harmonic field. Quantities are plotted in units of (ε0∣E02).

Fig. 5
Fig. 5

Energy index (using Eq. (31)) calculated with multi-Lorentz approach and when only Drude contribution is taken into account.

Fig. 6
Fig. 6

Metal-Dielectric-Metal waveguide.

Fig. 7
Fig. 7

Real and imaginary parts of the effective index neff as a function of angular frequency.

Fig. 8
Fig. 8

Group index Ng as a function of angular frequency for an Al-based MDM waveguide with w = 50 nm. The Al material model used is the Drude-Multi-Lorentz approach.

Fig. 9
Fig. 9

Average total stored energy density (left panel) and averaged dissipated power (right panel) obtained for the fundamental mode of the MDM waveguide at the Al spectral resonance (ω = 2.285 × 1015 rad/s) at x0 = 0.

Fig. 10
Fig. 10

Energy index for the fundamental mode of a MDM (Al-Glass-Al) waveguide with channel width w = 50 nm.

Tables (1)

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Table 1 Model Parameters for Al Permittivity

Equations (44)

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u s (t) = ε 0 | E 0 | 2 4 [ ε (ω)+ω d ε (ω) dω ],
u s = 1 4 ε 0 | E 0 | 2 ( ( ε 1)+ 2 ε ω γ )
W d = 1 2 ( ε ε 0 ω ) | E 0 | 2 ,
S= ε 0 E dE dt + μ 0 H dH dt +E dP dt .
P= j=1 M ρ j p j = j=1 M ρ j ( -e r j )
m e r ¨ j + m e γ r ˙ j + m e ω 0 r ˙ j =eE
E= m e e ( r ¨ j + γ j r ˙ j + ω 0j 2 r j ).
E dP dt = j=1 M ( m e ρ j )( r ¨ j + γ j r ˙ j + ω 0j 2 r j ) r ˙ j .
( r ¨ j r ˙ j + ω 0j 2 r j r ˙ j )= 1 2 t ( r ˙ j r ˙ j + ω 0j 2 r j r j ).
E dP dt ={ t j=1 M ρ j ( 1 2 m e ( r ˙ j r ˙ j )+ 1 2 m e ω 0j 2 ( r j r j ) ) + j=1 M m e ρ j j γ j ( r ˙ j r ˙ j ) }.
K j = 1 2 m e ( r ˙ j r ˙ j ) and U j = 1 2 m e ω 0j 2 ( r j r j ).
E dP dt ={ t j=1 M ρ j ( K j + U j ) + j=1 M ρ j γ j ( r ˙ j r ˙ j ) }.
S= t [ 1 2 ε o ( EE )+ 1 2 μ o ( HH )+ u s (t) ]+ W d (t),
u s (t)= j=1 M ρ j ( K j + U j ) and W d (t)= j=1 M ρ j γ j ( r ˙ j r ˙ j )
r j =[ -e / m e ( ω 0j 2 - ω 2 )-i( γ j ω ) ]E.
p j =e r j =[ e 2 / m 0 ε 0 ( ω 0j 2 - ω 2 )-i( γ j ω ) ] ε 0 E,
P= j=1 M ρ j e 2 / m e ε 0 ( ω 2 - ω 0j 2 )-i( γ j ω ) = χ e (ω) ε 0 E,
χ e (ω)= j=1 M ρ j e 2 / m e ε 0 ( ω 0j 2 - ω 2 )-i( γ j ω ) .
ε= ε +i ε =[ 1+ j=1 M f j ω p 2 ( ω 0j 2 - ω 2 )-i( γ j ω ) ]
ω p 2 =( ρ e 2 m e ε 0 )
ε =1+ j=1 M f j ω p 2 ( ω 2 - ω 0j 2 ) ( ω 0j 2 - ω 2 ) 2 + ( γ j ω ) 2
ε = j=1 M f j ω p 2 ( γ j ω ) ( ω 0j 2 - ω 2 ) 2 + ( γ j ω ) 2
r ˙ j 2 = ω 2 r j 2 and r ˙ j 2 + ω j 2 r j 2 =( ω 2 + ω j 2 ) r j 2 .
u s (t) = j=1 M ρ j ( K j + U j ) = 1 4 ε 0 | E | 2 [ ( ε 1 )+ j 2ω γ j ε j ].
W d (t) = j=1 M m n j γ j r ˙ j 2 = 1 2 ω ε ε 0 | E | 2 ,
u s (t) = 1 4 ε 0 | E | 2 [ ( ε 1 )+ 2ω ε j γ j Γ ].
ε γ j = j γ j 1 ε j j γ j 1 Γ= j γ j
ε (ω)=1- f 1 ω 2 ω p 2 ω 4 + ( γ 1 ω ) 2 + f 2 ( ω 02 2 - ω 2 ) ω p 2 ( ω 02 2 - ω 2 ) 2 + ( γ 2 ω ) 2 + f 3 ( ω 03 2 - ω 2 ) ω p 2 ( ω 03 2 - ω 2 ) 2 + ( γ 3 ω ) 2
ε (ω)= f 1 γ 1 ω ω p 2 ω 4 + ( γ 1 ω ) 2 + f 2 γ 2 ω ω p 2 ( ω 02 2 - ω 2 ) 2 + ( γ 2 ω ) 2 + f 3 γ 3 ω ω p 2 ( ω 03 2 - ω 2 ) 2 + ( γ 3 ω ) 2 ,
η= ε =n+iκ
N g =Re( η+ω dη dω ).
v E = S 1 2 ε o E E + 1 2 μ o H H + u s .
N E =[ n+ ω ε j γ j nΓ ].
2 H y z 2 -( ε ε 0 k 0 2 - β 2 ) H y =0
E x =i 1 ωε ε 0 H y z E z = β ωε ε 0 H y .
H y (z)=A{ cosh( k d w/2) e iβx e - k m (z-w/2) (w/2)<z< cosh( k d z) e iβx (-w/2)z(w/2) cosh( k d w/2) e iβx e - k m (z+w/2) -<z<(w/2)
k d,m = β 2 ε d,m k 0 2 .
tanh( k d w 2 )=( ε d k m ε m k d ),
v E = -D/2 D/2 dy - S x ( x 0 ,z) dz -D/2 D/2 dy - 1 2 ε o EE+ 1 2 μ o HH+ u s ( x 0 ,z) dz = P d + P m U em + U d + U m ,
P m =2 w/2 S mz dz = β 2 k m ω ε d (ω) ε 0 | A | 2
P d =2 0 w/2 S dz dz = β 4ω ε d (ω) ε 0 [ sinh( k d w) k d + sin( k d w) k d ] | A | 2
U em = -D/2 D/2 dy - 1 2 ε o EE+ 1 2 μ o HH dz
U m =2 w/2 u m dz =( 1 4 k m ){ μ 0 +[ ( ε 1 )+ j 2ω γ j ε j ] ( | β | 2 + | k m | 2 ) ω 2 | ε m | 2 ε 0 } | A | 2
U d =2 0 w/2 u d dz = w 8 { [ ( | k m |w ) 2 2 ω 2 ε d ε 0 ]+[ μ o + ( β 2 + | k m | 2 ) ω 2 ε d ε 0 ][ sinh( k d w) k d w ]+ [ μ o + ( | β | 2 | k m | 2 ) ω 2 ε d ε 0 ][ sin( k d w) k d w ] } | A | 2

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